Let $K^n$ be the vector space of rows of length $n$ over $K$. A subspace $S$ is called coordinately complete if for every $i\ (1 \leq i \leq n)$ there is an element in $S$ with a nonzero $i$th coordinate.
The number $V_q(n,t)$ of all coordinately complete $t$-dimensional subspaces of $K^n$ with $K=GF(q)$ is $$V_q(n,t) = \sum_{k=0}^{n-t}(-1)^k\binom{n}{k}{n-k \brack t}_q.$$ (Here ${n \brack k}_q$ is well-known $q$-binomial coefficient.)
Let $\text{S}_q[n,t]$ be the $q$-Stirling number of the second kind. Today I've found that $$V_q(n,t) = \frac{(q-1)^{n-t}}{q^{t(t-1)/2}}\text{S}_q[n,t].$$
The name of coordinately complete subspaces was invented by the authors (including me) of the recent paper [1]. After finding connection with $q$-Stirling numbers I suspect this subspaces might have a proper name in existing literature. I will appreciate any reference.
- Enveloping algebras and ideals of the niltriangular subalgebra of the Chevalley algebra. G. Egorychev et al. Siberian Mathematical Journal, 2023, Vol. 64, No. 2, pp. 300–317.